problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Find $1-0.\overline{9}.$ | 0 |
Bruce's aunt gave him $71 to spend on clothes at the mall. He bought 5 shirts that cost $5 each and a pair of pants that cost $26. How much money does Bruce have left to buy more clothes? | Bruce spent 5 * $5 on shirts = $<<5*5=25>>25.
Including the pants, Bruce spent $25 + $26 = $<<25+26=51>>51.
Since he began with $71, he has $71 - $51 spent = $<<71-51=20>>20 remaining.
#### 20 |
Four elevators in a skyscraper, differing in color (red, blue, green, and yellow), move in different directions at different but constant speeds. An observer timed the events as follows: At the 36th second, the red elevator caught up with the blue one (moving in the same direction). At the 42nd second, the red elevator... | 46 |
Compute the value of \(1^{25}+2^{24}+3^{23}+\ldots+24^{2}+25^{1}\). | 66071772829247409 |
Given positive numbers \(a, b, c, x, y, z\) that satisfy the equations \(cy + bz = a\), \(az + cx = b\), and \(bx + ay = c\), find the minimum value of the function
\[ f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}. \] | 1/2 |
Sofia ran $5$ laps around the $400$-meter track at her school. For each lap, she ran the first $100$ meters at an average speed of $4$ meters per second and the remaining $300$ meters at an average speed of $5$ meters per second. How much time did Sofia take running the $5$ laps? | 7 minutes and 5 seconds |
In the diagram, $AB$ is parallel to $DC,$ and $ACE$ is a straight line. What is the value of $x?$ [asy]
draw((0,0)--(-.5,5)--(8,5)--(6.5,0)--cycle);
draw((-.5,5)--(8.5,-10/7));
label("$A$",(-.5,5),W);
label("$B$",(8,5),E);
label("$C$",(6.5,0),S);
label("$D$",(0,0),SW);
label("$E$",(8.5,-10/7),S);
draw((2,0)--(3,0),Arro... | 35 |
Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge. (12 points) | \frac{9\sqrt{3}}{4} |
What integer $n$ satisfies $0 \leq n < 151$ and $$100n \equiv 93 \pmod {151}~?$$ | 29 |
How many different 7-digit positive integers exist? (Note that we don't allow "7-digit" integers that start with 0, such as 0123456; this is actually a 6-digit integer.) | 9,\!000,\!000 |
2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon? | \dfrac{14}{39} |
The ratio of cows to bulls in a barn is 10:27. If there are 555 cattle on the farm, how many bulls are on the farm? | The total ratio of cattle on the farm is 10+27 = <<10+27=37>>37
Since the fraction that represents bulls from the total ratio is 27/37, the number of bulls on the farm is 27/37*555 = <<27/37*555=405>>405
#### 405 |
How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? | 4 |
A Senate committee consists of 10 Republicans and 8 Democrats. In how many ways can we form a subcommittee that has at most 5 members, including exactly 3 Republicans and at least 2 Democrats? | 10080 |
There are two hourglasses - one for 7 minutes and one for 11 minutes. An egg needs to be boiled for 15 minutes. How can you measure this amount of time using the hourglasses? | 15 |
Given a right triangle \( ABC \) with hypotenuse \( AB \). One leg \( AC = 15 \) and the altitude from \( C \) to \( AB \) divides \( AB \) into segments \( AH \) and \( HB \) with \( HB = 16 \). What is the area of triangle \( ABC \)? | 150 |
Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(... | 50 |
In the Cartesian coordinate plane \( xOy \), the coordinates of point \( F \) are \((1,0)\), and points \( A \) and \( B \) lie on the parabola \( y^2 = 4x \). It is given that \( \overrightarrow{OA} \cdot \overrightarrow{OB} = -4 \) and \( |\overrightarrow{FA}| - |\overrightarrow{FB}| = 4\sqrt{3} \). Find the value of... | -11 |
A rectangular pasture is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 15 meters along the fence including at the points where the fence meets the rock wall. Given the dimensions of the pasture are 36 m by 75 m, find the minimum number of post... | 14 |
Jenna is at a fair with four friends. They all want to ride the roller coaster, but only three people can fit in a car. How many different groups of three can the five of them make? | 10 |
The line $2x+ay-2=0$ is parallel to the line $ax+(a+4)y-4=0$. Find the value of $a$. | -2 |
Molly and her parents love to hang out at the beach on weekends. Molly spent the better part of the day at the beach and saw 100 people join them. At 5:00, 40 people left the beach. What's the total number of people at the beach if they all stayed until evening? | Molly and her parent were 2+1 = <<2+1=3>>3 people.
When 100 more people joined, the total number became 3+100 = <<3+100=103>>103
If 40 people left before evening, the number reduced to 103-40 = 63 people
#### 63 |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they d... | 25\% |
Let \[g(x) =
\begin{cases}
2x - 4 &\text{if } x < 0, \\
5 - 3x &\text{if } x \geq 0.
\end{cases}\]
Find $g(-2)$ and $g(3)$. | -4 |
Given the function \( y = \frac{1}{2}\left(x^{2}-100x+196+\left|x^{2}-100x+196\right|\right) \), calculate the sum of the function values when the variable \( x \) takes on the 100 natural numbers \( 1, 2, 3, \ldots, 100 \). | 390 |
A high school basketball team has 16 players, including a set of twins named Bob and Bill, and another set of twins named Tom and Tim. How many ways can we choose 5 starters if no more than one player from each set of twins can be chosen? | 3652 |
Christine must buy at least $45$ fluid ounces of milk at the store. The store only sells milk in $200$ milliliter bottles. If there are $33.8$ fluid ounces in $1$ liter, then what is the smallest number of bottles that Christine could buy? (You may use a calculator on this problem.) | 7 |
A geometric sequence of positive integers is formed for which the first term is 2 and the fifth term is 162. What is the sixth term of the sequence? | 486 |
The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? | 36 |
When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes: | -1 |
Given the parametric equation of line $l$ as $$\begin{cases} x=t \\ y= \frac { \sqrt {2}}{2}+ \sqrt {3}t \end{cases}$$ (where $t$ is the parameter), if the origin $O$ of the Cartesian coordinate system $xOy$ is taken as the pole and the direction of $Ox$ as the polar axis, and the same unit of length is chosen to esta... | \frac { \sqrt {10}}{2} |
A notebook with 75 pages numbered from 1 to 75 is renumbered in reverse, from 75 to 1. Determine how many pages have the same units digit in both the old and new numbering systems. | 15 |
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal equals 7, and the sum on the other diagonal equals 21. What is the sum of the numbers in the five shaded cells? | 25 |
Two circles centered at \( O_{1} \) and \( O_{2} \) have radii 2 and 3 and are externally tangent at \( P \). The common external tangent of the two circles intersects the line \( O_{1} O_{2} \) at \( Q \). What is the length of \( PQ \)? | 12 |
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:
- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.
... | 15 |
Selena reads a book with 400 pages. Harry reads a book with 20 fewer than half the number of pages Selena's book has. How many pages are in the book of Harry? | Half the number of pages of Selena is 400/2 = <<400/2=200>>200.
Therefore, Harry's book has 200 - 20 = <<200-20=180>>180 pages.
#### 180 |
Compute $\arccos \frac{\sqrt{3}}{2}.$ Express your answer in radians. | \frac{\pi}{6} |
The fifth grade has 120 teachers and students going to visit the Natural History Museum. A transportation company offers two types of vehicles to choose from:
(1) A bus with a capacity of 40 people, with a ticket price of 5 yuan per person. If the bus is full, the ticket price can be discounted by 20%.
(2) A minivan wi... | 480 |
In a sequence of triangles, each successive triangle has its small triangles numbering as square numbers (1, 4, 9,...). Each triangle's smallest sub-triangles are shaded according to a pascal triangle arrangement. What fraction of the eighth triangle in the sequence will be shaded if colors alternate in levels of the p... | \frac{1}{4} |
Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$ | 259 |
Given a cone with vertex $S$, and generatrices $SA$, $SB$ perpendicular to each other, and the angle between $SA$ and the base of the cone is $30^{\circ}$. If the area of $\triangle SAB$ is $8$, then the volume of this cone is ______. | 8\pi |
Let $a,$ $b,$ $c,$ $d,$ $e$ be positive real numbers such that $a^2 + b^2 + c^2 + d^2 + e^2 = 100.$ Let $N$ be the maximum value of
\[ac + 3bc + 4cd + 8ce,\]and let $a_N,$ $b_N$, $c_N,$ $d_N,$ $e_N$ be the values of $a,$ $b,$ $c,$ $d,$ $e,$ respectively, that produce the maximum value of $N.$ Find $N + a_N + b_N + c_... | 16 + 150\sqrt{10} + 5\sqrt{2} |
Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: "The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 points, the machine gives... | 11 |
Given the sequence $\{a_n\}$ such that the sum of the first $n$ terms is $S_n$, $S_1=6$, $S_2=4$, $S_n>0$ and $S_{2n}$, $S_{2n-1}$, $S_{2n+2}$ form a geometric progression, while $S_{2n-1}$, $S_{2n+2}$, $S_{2n+1}$ form an arithmetic progression. Determine the value of $a_{2016}$.
Options:
A) $-1009$
B) $-1008$
C) $-100... | -1009 |
Find all values of $x$ that satisfy the equation $|x-3|=2x+4$. Express your answers in simplest fractional form. | -\frac13 |
Each positive integer number $n \ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$ , $p_ {32} = 6$ , $p_ {203} = 6$ . Let $S = p_1 + p_2 + p_3 + \dots + p_ {999}$ . Find the largest prime that divides $S $ . | 103 |
Find $x$, given that $x$ is neither zero nor one and the numbers $\{x\}$, $\lfloor x \rfloor$, and $x$ form a geometric sequence in that order. (Recall that $\{x\} = x - \lfloor x\rfloor$). | 1.618 |
What is the smallest square number that, when divided by a cube number, results in a fraction in its simplest form where the numerator is a cube number (other than 1) and the denominator is a square number (other than 1)? | 64 |
What is the maximum value that can be taken by the sum
$$
\left|x_{1}-1\right|+\left|x_{2}-2\right|+\ldots+\left|x_{63}-63\right|
$$
if $x_{1}, x_{2}, \ldots, x_{63}$ are some permutation of the numbers $1, 2, 3, \ldots, 63$? | 1984 |
The Cubs are playing the Red Sox in the World Series. To win the world series, a team must win 4 games before the other team does. If the Cubs win each game with probability $\dfrac{3}{5}$ and there are no ties, what is the probability that the Cubs will win the World Series? Express your answer as a percent rounded to... | 71 |
Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible? | 480 |
Find the greatest common divisor of 91 and 72. | 1 |
The graph of \[y^4 - 4x^4 = 2y^2 - 1\]is the union of the graphs of two different conic sections. Which two types of conic sections are they?
(Write your answer as a list, with "C" for circle, "E" for ellipse, "H" for hyperbola, and "P" for parabola. For example, "C, H" if you think the graph consists of a circle and ... | \text{H, E} |
Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi}{2} - 9 |
Three of the four vertices of a rectangle are \((1, 7)\), \((14, 7)\), and \((14, -4)\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \((x - 1)^2 + (y + 4)^2 = 16\)? | 4\pi |
Let \([x]\) represent the integral part of the decimal number \(x\). Given that \([3.126] + \left[3.126 + \frac{1}{8}\right] + \left[3.126 + \frac{2}{8}\right] + \ldots + \left[3.126 + \frac{7}{8}\right] = P\), find the value of \(P\). | 25 |
$3^n = 3 \cdot 9^3 \cdot 81^2$. What is the value of $n$? | 15 |
The volume of a given sphere is $36\pi$ cubic inches. How many square inches are in its surface area? Express your answer in terms of $\pi$. | 36\pi |
Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167 y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\). | 2020032 |
Mr. Isaac rides his bicycle at the rate of 10 miles per hour for 30 minutes. If he rides for another 15 miles, rests for 30 minutes, and then covers the remaining distance of 20 miles, what's the total time in minutes took to travel the whole journey? | If in one hour he covers 10 miles and rides for 15 miles at the rate of 10 miles per hour, then the time taken for the 15 miles will be 15/10 = <<15/10=1.5>>1.5 hours.
The time in minutes is 1.5 * 60 = <<1.5*60=90>>90 minutes.
The time taken for the remaining 20 miles at a rate of 10 miles per hour is 20/10= <<20/10=2>... |
In the diagram, $QRS$ is a straight line. What is the measure of $\angle RPS,$ in degrees? [asy]
pair Q=(0,0);
pair R=(1.3,0);
pair SS=(2.3,0);
pair P=(.8,1);
draw(P--Q--R--SS--P--R);
label("$Q$",Q,S);
label("$R$",R,S);
label("$S$",SS,S);
label("$P$",P,N);
label("$48^\circ$",Q+(.12,.05),NE);
label("$67^\circ$",P-(.0... | 27^\circ |
Colby harvested his mango trees, the total mangoes he harvested is 60 kilograms. He sold 20 kilograms to the market and sold the remaining half to his community. If each kilogram contains 8 mangoes, how many mangoes does he still have? | He has 60-20= <<60-20=40>>40 kilograms of mangoes left after selling them to the market.
Colby sold 1/2 x 40 = <<1/2*40=20>>20 kilograms of mangoes to the community.
Therefore, Colby still has 20x8= <<20*8=160>>160 pieces of mangoes.
#### 160 |
Milly's babysitter charges $16/hour. Milly is considering switching to a new babysitter who charges $12/hour, but also charges an extra $3 for each time the kids scream at her. If Milly usually hires the babysitter for 6 hours, and her kids usually scream twice per babysitting gig, how much less will the new babysitter... | First figure out how much the first babysitter costs by multiplying her hourly rate by the number of hours she works: $16/hour * 6 hours = $<<16*6=96>>96
Then use the same method to figure out how much the second babysitter charges before any screaming: $12/hour * 6 hours = $<<12*6=72>>72
Then figure out how much the b... |
Express as a common fraction in simplest form: $$
\sqrt{6\frac{1}{4}}
$$ | \frac{5}{2} |
Given a student measures his steps from one sidewalk to another on a number line where each marking represents 3 meters, calculate the position marking $z$ after taking 5 steps from the starting point, given that the student counts 8 steps and the total distance covered is 48 meters. | 30 |
In a particular sequence, the first term is $a_1 = 1009$ and the second term is $a_2 = 1010$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = 2n$ for all $n \ge 1$. Determine $a_{1000}$. | 1675 |
Given two similar triangles $\triangle ABC\sim\triangle FGH$, where $BC = 24 \text{ cm}$ and $FG = 15 \text{ cm}$. If the length of $AC$ is $18 \text{ cm}$, find the length of $GH$. Express your answer as a decimal to the nearest tenth. | 11.3 |
Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then: | 1 |
What is the least positive integer that has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4? | 10 |
Circles $P$, $Q$, and $R$ are externally tangent to each other and internally tangent to circle $S$. Circles $Q$ and $R$ are congruent. Circle $P$ has radius 2 and passes through the center of $S$. What is the radius of circle $Q$? | \frac{16}{9} |
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$ . What is the area of $R$ divided by the area of $ABCDEF$ ? | 1/3 |
Let $\mathbf{a}$ and $\mathbf{b}$ be nonzero vectors such that
\[\|\mathbf{a}\| = \|\mathbf{b}\| = \|\mathbf{a} + \mathbf{b}\|.\]Find the angle between $\mathbf{a}$ and $\mathbf{b},$ in degrees. | 120^\circ |
Given in $\bigtriangleup ABC$, $AB = 75$, and $AC = 120$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. Find the length of $BC$. | 117 |
A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle? | 2.37 |
Solve the equation $a^3 + b^3 + c^3 = 2001$ in positive integers. | \[
\boxed{(10,10,1), (10,1,10), (1,10,10)}
\] |
Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square. | 35 |
Determine the value of the infinite product $(3^{1/4})(9^{1/16})(27^{1/64})(81^{1/256}) \dotsm$ plus 2, the result in the form of "$\sqrt[a]{b}$ plus $c$". | \sqrt[9]{81} + 2 |
Suppose that $P(z)$, $Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$. What is the minimum possible value... | 1 |
Suppose \( x \), \( y \), and \( z \) are positive numbers satisfying:
\[
x^2 \cdot y = 2, \\
y^2 \cdot z = 4, \text{ and} \\
z^2 / x = 5.
\]
Find \( x \). | 5^{1/7} |
If $P = \sqrt{1988 \cdot 1989 \cdot 1990 \cdot 1991 + 1} + \left(-1989^{2}\right)$, calculate the value of $P$. | 1988 |
Given that the graph of a power function passes through the points $(2, 16)$ and $\left( \frac{1}{2}, m \right)$, then $m = \_\_\_\_\_\_$. | \frac{1}{16} |
How many even integers are there between \( \frac{12}{3} \) and \( \frac{50}{2} \)? | 10 |
What is the greatest possible sum of two consecutive integers whose product is less than 400? | 39 |
Let $m=\underbrace{22222222}_{\text{8 digits}}$ and $n=\underbrace{444444444}_{\text{9 digits}}$.
What is $\gcd(m,n)$? | 2 |
Simplify first, then evaluate: $(\frac{{x-1}}{{x-3}}-\frac{{x+1}}{x})÷\frac{{{x^2}+3x}}{{{x^2}-6x+9}}$, where $x$ satisfies $x^{2}+2x-6=0$. | -\frac{1}{2} |
Consider the set of all points $(x,y)$ in the coordinate plane for which one of the coordinates is exactly twice the other. If we were to plot all such points, into how many regions would the resulting graph split up the plane? | 4 |
Our club has 20 members, 10 boys and 10 girls. In how many ways can we choose a president and a vice-president if they must be of different gender? | 200 |
What fraction of the area of an isosceles trapezoid $KLMN (KL \parallel MN)$ is the area of triangle $ABC$, where $A$ is the midpoint of base $KL$, $B$ is the midpoint of base $MN$, and $C$ is the midpoint of leg $KN$? | \frac{1}{4} |
The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r$ is $\frac 27.$ What is the number of pos... | 417 |
I have created a new game where for each day in May, if the date is a prime number, I walk three steps forward; if the date is composite, I walk one step backward. If I stop on May 31st, how many steps long is my walk back to the starting point? | 14 |
Tom decides to get a new floor for his room. It cost $50 to remove the floor. The new floor costs $1.25 per square foot and Tom's room is 8*7 feet. How much did it cost to replace the floor? | The room is 8*7=<<8*7=56>>56 square feet
So the new carpet cost 56*1.25=$<<56*1.25=70>>70
So the total cost was 50+70=$<<50+70=120>>120
#### 120 |
There is a magical tree with 58 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if on any given day the number of fruits on the tree is less than the number of fruits that should fall on that ... | 12 |
$100_{10}$ in base $b$ has exactly $5$ digits. What is the value of $b$? | 3 |
Three friends agreed to pay an equal amount of money for buying 5 bags of chips. If they each pay $5, how much does each bag of chips cost? | Five bags of chips cost $5 x 3 = $<<5*3=15>>15.
So each bag of chips costs $15/5 = $<<15/5=3>>3.
#### 3 |
Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta, who did not att... | \frac{349}{500} |
There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits that should fall on a given day, the fall... | 17 |
A sequence is defined as follows: $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}= 6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum_{k=1}^{28}a_k$ is divided by 1000. | 834 |
Let $ABCD$ be a parallelogram and let $\overrightarrow{AA^\prime}$, $\overrightarrow{BB^\prime}$, $\overrightarrow{CC^\prime}$, and $\overrightarrow{DD^\prime}$ be parallel rays in space on the same side of the plane determined by $ABCD$. If $AA^{\prime} = 10$, $BB^{\prime}= 8$, $CC^\prime = 18$, and $DD^\prime = 22$ a... | 1 |
Stormi is saving up to buy a bicycle. She washes 3 cars for $10 each. She mows 2 lawns for $13 each. If the bicycle she wants costs $80, how much more money does Stormi need to make to afford the bicycle? | The total amount of money from washing cars is 3 * $10 = $<<3*10=30>>30
The total amount of money from mowing lawns 2 * $13 = $<<2*13=26>>26
The total amount of money Stormi makes is $30 + $26 = $<<30+26=56>>56
Stormi still needs to make $80 - $56 = $<<80-56=24>>24
#### 24 |
Given the vertex of angle α is at the origin of the coordinate system, its initial side coincides with the non-negative half-axis of the x-axis, and its terminal side passes through the point (-√3,2), find the value of tan(α - π/6). | -3\sqrt{3} |
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